Classically, $S_n$ occurs as a Galois group for
certain $x^n+ax+b$, $n\geq 5$. That means that
obstructions for $p$ splitting over such $L$
must reflect information beyond the Galois group
of $p$. So absent a full answer to my question,
what candidates does one have for such an obstruction?
For example, does the form of the polynomial single
out particular representations of $S_n$?

Again, absent a full answer, does the literature contain theorems about
polynomials not splitting over similar large extension of ${\Bbb Q}$?

1 Answer
1

A minor note: it suffices to limit oneself to closure under roots of polynomials of form $x^n + b$ or $x^n + x + b$, since any other polynomial of form $x^n + ax + b$ can be transformed into the latter by the change of variables $x = a^{1/(n-1)}y$ (and $a^{1/(n-1)}$ is 'available' by virtue of the former).